Lemma 11 ([1]}) 

Assume that \(X\)  is a random variable defined in a measurable space \(\left(\mathcal {X},\mathcal {A}\right)\) , and \(\left(\Theta ,\mathcal {F}\right)\)  is a measurable parameter space.
Let \(\mu \)  be a random measure on \(\left(\Theta ,\mathcal {F}\right)\)  and \(h:\mathcal {X}\times \Theta \rightarrow \mathbb {R}\)  be a real-valued \(\mathcal {A}\otimes \mathcal {F}/\mathcal {B}\left(\mathbb {R}\right)\) -measurable function such that \(\mathbb {E}_{X}[\exp \left(h\left(X,\theta \right)\right)]<\infty \)  for \(\mu \) -almost all \(\theta \) .
It holds that with probability at least \(1-e^{-t}\) , for all the probability measures \(\rho \ll \mu \)  simultaneously,
\(\mathbb {E}_{\rho }[h\left(X,\theta \right)]\le \mathbb {E}_{\rho }\left[\log \mathbb {E}_{X}\left[\exp \left(h\left(X,\theta \right)\right)\right]\right]+ \mathrm {KL}\left(\rho \Vert  \mu \right)+t.\) 

